# Contents

Preface
Definitions and miscellaneous formulas
0.1
Introduction
0.2
The q-shifted factorials
0.3
The q-gamma function and the q-binomial coefficient
0.4
Hypergeometric and basic hypergeometric functions
0.5
The q-binomial theorem and other summation formulas
0.6
Transformation formulas
0.7
Some special functions and their q-analogues
0.8
The q-derivative and the q-integral
0.9
Shift operators and Rodrigues-type formulas
1
Hypergeometric orthogonal polynomials
1.1
Wilson
1.2
Racah
1.3
Continuous dual Hahn
1.4
Continuous Hahn
1.5
Hahn
1.6
Dual Hahn
1.7
Meixner-Pollaczek
1.8
Jacobi
1.8.1
Gegenbauer / Ultraspherical
1.8.2
Chebyshev
1.8.3
Legendre / Spherical
1.9
Meixner
1.10
Krawtchouk
1.11
Laguerre
1.12
Charlier
1.13
Hermite
2
Limit relations between hypergeometric orthogonal polynomials
2.1
Wilson
2.2
Racah
2.3
Continuous dual Hahn
2.4
Continuous Hahn
2.5
Hahn
2.6
Dual Hahn
2.7
Meixner-Pollaczek
2.8
Jacobi
2.8.1
Gegenbauer / Ultraspherical
2.9
Meixner
2.10
Krawtchouk
2.11
Laguerre
2.12
Charlier
2.13
Hermite
q-SCHEME
3
Basic hypergeometric orthogonal polynomials
3.1
3.2
q-Racah
3.3
Continuous dual q-Hahn
3.4
Continuous q-Hahn
3.5
Big q-Jacobi
3.5.1
Big q-Legendre
3.6
q-Hahn
3.7
Dual q-Hahn
3.8
Al-Salam-Chihara
3.9
q-Meixner-Pollaczek
3.10
Continuous q-Jacobi
3.10.1
Continuous q-ultraspherical / Rogers
3.10.2
Continuous q-Legendre
3.11
Big q-Laguerre
3.12
Little q-Jacobi
3.12.1
Little q-Legendre
3.13
q-Meixner
3.14
Quantum q-Krawtchouk
3.15
q-Krawtchouk
3.16
Affine q-Krawtchouk
3.17
Dual q-Krawtchouk
3.18
Continuous big q-Hermite
3.19
Continuous q-Laguerre
3.20
Little q-Laguerre / Wall
3.21
q-Laguerre
3.22
Alternative q-Charlier
3.23
q-Charlier
3.24
Al-Salam-Carlitz I
3.25
Al-Salam-Carlitz II
3.26
Continuous q-Hermite
3.27
Stieltjes-Wigert
3.28
Discrete q-Hermite I
3.29
Discrete q-Hermite II
4
Limit relations between basic hypergeometric orthogonal polynomials
4.1
4.2
q-Racah
4.3
Continuous dual q-Hahn
4.4
Continuous q-Hahn
4.5
Big q-Jacobi
4.6
q-Hahn
4.7
Dual q-Hahn
4.8
Al-Salam-Chihara
4.9
q-Meixner-Pollaczek
4.10
Continuous q-Jacobi
4.10.1
Continuous q-ultraspherical / Rogers
4.11
Big q-Laguerre
4.12
Little q-Jacobi
4.13
q-Meixner
4.14
Quantum q-Krawtchouk
4.15
q-Krawtchouk
4.16
Affine q-Krawtchouk
4.17
Dual q-Krawtchouk
4.18
Continuous big q-Hermite
4.19
Continuous q-Laguerre
4.20
Little q-Laguerre / Wall
4.21
q-Laguerre
4.22
Alternative q-Charlier
4.23
q-Charlier
4.24
Al-Salam-Carlitz I
4.25
Al-Salam-Carlitz II
4.26
Continuous q-Hermite
4.27
Stieltjes-Wigert
4.28
Discrete q-Hermite I
4.29
Discrete q-Hermite II
5
From basic to classical hypergeometric orthogonal polynomials
5.1
5.2
q-Racah --> Racah
5.3
Continuous dual q-Hahn --> Continuous dual Hahn
5.4
Continuous q-Hahn --> Continuous Hahn
5.5
Big q-Jacobi --> Jacobi
5.5.1
Big q-Legendre --> Legendre / Spherical
5.6
q-Hahn --> Hahn
5.7
Dual q-Hahn --> Dual Hahn
5.8
Al-Salam-Chihara --> Meixner-Pollaczek
5.9
q-Meixner-Pollaczek --> Meixner-Pollaczek
5.10
Continuous q-Jacobi --> Jacobi
5.10.1
Continuous q-ultraspherical / Rogers --> Gegenbauer / Ultraspherical
5.10.2
Continuous q-Legendre --> Legendre / Spherical
5.11
Big q-Laguerre --> Laguerre
5.12
Little q-Jacobi --> Jacobi / Laguerre
5.12.1
Little q-Legendre --> Legendre / Spherical
5.13
q-Meixner --> Meixner
5.14
Quantum q-Krawtchouk --> Krawtchouk
5.15
q-Krawtchouk --> Krawtchouk
5.16
Affine q-Krawtchouk --> Krawtchouk
5.17
Dual q-Krawtchouk --> Krawtchouk
5.18
Continuous big q-Hermite --> Hermite
5.19
Continuous q-Laguerre --> Laguerre
5.20
Little q-Laguerre / Wall --> Laguerre / Charlier
5.21
q-Laguerre --> Laguerre / Charlier
5.22
Alternative q-Charlier --> Charlier
5.23
q-Charlier --> Charlier
5.24
Al-Salam-Carlitz I --> Charlier / Hermite
5.25
Al-Salam-Carlitz II --> Charlier / Hermite
5.26
Continuous q-Hermite --> Hermite
5.27
Stieltjes-Wigert --> Hermite
5.28
Discrete q-Hermite I --> Hermite
5.29
Discrete q-Hermite II --> Hermite
Bibliography
Index